Hereditary properties of character injectivity with applications to semigroup algebras

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by  M. Essmaili

HEREDITARY PROPERTIES OF CHARACTER INJECTIVITY
WITH APPLICATIONS TO SEMIGROUP ALGEBRAS
∗
M. ESSMAILI , M. FOZOUNI AND J. LAALI
5 Abstract. In this paper, we investigate the notion φ-injectivity for Banach
1 A-modules,whereφisacharacteronA.We obtainsomehereditaryproperties
0 of φ-injectivity for certain classes of Banach modules related to closed ideals.
2
These results allow us to study φ-injectivity of certain Banach A-modules in

n commutativecase,speciallyℓ1-semilatticealgebras. Asanapplication,wegive
a
an example of a non-injective Banach module which is φ-injective for each
J
character φ.
3

]
A
F 1. introduction
.
h
Suppose that A is a Banach algebra. We denote by A-mod and mod-A the
t
a
categoriesofBanachleftA-modulesandBanachrightA-modules, respectively. In
m
the case where A is unital, we also denote by A-unmod the categories of unital
[
Banach left A-modules. For each E,F ∈ A-mod, let B(E,F) be the closed
A
1
subspace of B(E,F) consisting of the left A-module morphisms. An operator
v
5 T ∈ B(E,F) is called admissible if kerT and ImT are closed complemented
3 subspaces of E and F, respectively. It is easy to verify that T is admissible if and
5
only if there exists S ∈ B(F,E) such that T ◦S ◦T = T.
0
0 ABanach left A-module E is called injectiveif for each F,K ∈ A-mod and ad-
.
1 missible monomorphism T ∈ B(F,K), the induced map B(K,E) −→ B(F,E)
A A A
0 is onto. We also say E ∈ mod-A is flat if the dual module of E∗ ∈ A-mod is
5
1 injective with the following left module action:
:
v (a·f)(x) = f(x·a) (a ∈ A,x ∈ E).
i
X
The notions of injectivity and flatness of Banach algebras were introduced by A.
r
a Ya. Helemskii. These notions have been studied for various classes of Banach
modules; see [3], [6], [11] and [13] for more details. Recently, Ramsden in [11]
studied injectivity and flatness of Banach modules over semigroup algebras. It is
well known that if A is amenable, then every Banach A-modules is flat but the
converse is a long standing open problem. We recall that the answer is positive
for some classes of Banach algebras associated with locally compact groups such
as, the class of group algebras and measure algebras; see [3] and [12].
Kaniuth, Lau and Pym introduced and studied in [7] and [8] the notion of
φ-amenability for Banach algebras, where φ : A −→ C is a character, i.e., a non-
zero homomorphism on A. Afterwards, Monfared introduced and studied in [9]
the notion of character amenability for Banach algebras. Let ∆(A) be the set of
2010 Mathematics Subject Classification. Primary 46M10 Secondary 43A20, 46H25.
Key words and phrases. Injectivity, φ-injectivity, φ-amenability, semigroup algebras.
1
2 M. ESSMAILI,M. FOZOUNIANDJ. LAALI
all characters of the Banach algebra A, and let φ ∈ ∆(A). The Banach algebra
A is called left φ-amenable if for all Banach A-bimodules E for which the left
module action is given by
a·x = φ(a)x (a ∈ A,x ∈ E),
every derivation D : A −→ E∗ is inner. It is clear that amenability of A implies
φ-amenability for all φ ∈ ∆(A).
Recently, Nasr-Isfahani and Soltani Renani in [10] introduced and studied the
notion of φ-injectivity and φ-flatness for Banach modules (see Definition 2.1). As
an important result, it is shown in [10, Proposition 3.1] that the Banach algebra
A is left φ-amenable if and only if every Banach left A-modules E is φ-flat. In-
deed, this result gives a positive answer to the above open problem arises by A.
Ya. Helemskii in this homology setting based on character φ. Furthermore, they
obtained some necessary and sufficient conditions for φ-injectivity and character-
ized φ-injectivity of Banach modules in terms of a coretraction problem; see [10,
Theorem 2.4].
Thispaper isorganizedasfollows. InSection 2, afterrecalling some definitions,
we investigate some properties of φ-injectivity for Banach modules. Indeed, we
obtainasufficient conditionforφ-injectivityofBanachleftA-modules, inthecase
where A is a commutative Banach algebra. Moreover, we give some hereditary
properties of φ-injectivity for Banach A-modules related to the closed ideals of
Banach algebra A. As the main result, we show that if J is a left invariant
complemented ideal in A, then φ-injectivity of J andA/J inA-mod isequivalent
to the φ-injectivity of A in A-mod (Theorem 2.10). In Section 3, by using the
results of Section 2, we study φ-injectivity of certain ℓ1-semilattice algebras and
show that ℓ1(N ) as a Banach left ℓ1(N )-module is φ-injective for each character
∧ ∧
φ, although is not injective.
2. φ-injectivity and some hereditary properties
First, werecallsomestandardnotationsthatweshalluseanddefinethenotions
of φ-injectivity and φ-flatness of Banach modules.
Let A be a Banach algebra and E ∈ A-mod. Throughout the paper, we regard
E asaBanachleftA♯-module(theunitizationofA)withthefollowingleftmodule
action:
(a,λ)·x = a·x+λx (a ∈ A,λ ∈ C,x ∈ E).
Moreover, the space B(A,E) is a Banach A-bimodule with the following module
actions:
(a·T)(b) = T(ba), (T ·a)(b) = T(ab) (T ∈ B(A,E),a,b ∈ A).
Suppose that A is a Banach algebra and φ ∈ ∆(A). For each E ∈ A-mod we
define,
I(φ,E) = span{a·x−φ(a)x : a ∈ A,x ∈ E}.
Following [10], we also consider
B(A♯,E) = {T ∈ B(A♯,E) : T(ab−φ(b)a) = a·T(b−φ(b)e♯) for all a,b ∈ A},
φ
HEREDITARY PROPERTIES OF CHARACTER INJECTIVITY 3
where e♯ = (0,1) denotes the unite of A♯. It is straightforward to check that
B(A♯,E)isaclosed A-submoduleofB(A♯,E).Moreover, we definethe canonical
φ
morphism Π♯ : E −→ B(A♯,E) as follows:
φ φ
Π♯(x)(a) = a·x (x ∈ E,a ∈ A♯).
φ
Definition 2.1. Let A be a Banach algebra, φ ∈ ∆(A) and E ∈ A-mod. We say
that E is φ-injective if, for each F,K ∈ A-mod and admissible monomorphism
T : F −→ K with I(φ,K) ⊆ Im(T), the induced map T : B(K,E) −→
E A
B(F,E) defined by T (R) = R◦T is onto.
A E
The following theorem gives a characterization of φ-injectivity in terms of a
coretraction problem.
Theorem 2.2. ([10, Theorem 2.4]) Let A be a Banach algebra and φ ∈ ∆(A).
For E ∈ A-mod the following statements are equivalent.
(i) E is φ-injective.
(ii) Π♯ ∈ B(E, B(A♯,E)) is a coretraction, (that is there exists ρ♯ ∈
φ A φ φ
B( B(A♯,E),E) such that is a left inverse for Π♯).
A φ φ
A Banach right (left) A-module E is φ-flat if E∗ is φ-injective as a left (right)
A-module. It is shown that Banach algebra A is left φ-amenable if and only if
each Banach left A-module E is φ-flat [10, Proposition 3.1].
In this section, we give some hereditary properties of φ-injectivity for certain
classes of Banach modules. We also consider some hereditary properties of φ-
injectivity of Banach left A-modules with their ideals. We first give a sufficient
condition for φ-injectivity of Banach left A-module E in the case where A is a
commutative Banach algebra. Following [1, Definition 1.4.4], the annihilator of
E is defined by E⊥ = {a ∈ A : a·E = {0}}.
Theorem 2.3. Let A be a commutative Banach algebra, φ ∈ ∆(A) and E ∈
A-mod. If E⊥ ∩(ker(φ))c 6= ∅, then E is φ-injective.
Proof. Let a ∈ E⊥ ∩ (ker(φ))c. We can assume that φ(a ) = 1 and define the
0 0
map ρ♯ : B(A♯,E) −→ E by
φ φ
ρ♯(T) = T(e♯ −a ) (T ∈ B(A♯,E)).
φ 0 φ
Hence, for each x ∈ E we have
ρ♯◦ Π♯(x) = Π♯(x)(e♯ −a ) = (e♯ −a )·x = x.
φ φ φ 0 0
Therefore, ρ♯◦ Π♯ = I . On the other hand, for each a ∈ A and T ∈ B(A♯,E)
φ φ E φ
we have
ρ♯(a·T) = (a·T)(e♯ −a ) = T((e♯ −a )·a)
φ 0 0
= T(a−a a) (2.1)
0
= T(aφ(a )−aa ).
0 0
Since T ∈ B(A♯,E) we have T(aφ(a )−aa ) = a·T(e♯ −a ). Now, using (2.1)
φ 0 0 0
we conclude that
ρ♯(a·T) = a·T(e♯ −a ) = a· ρ♯(T).
φ 0 φ
4 M. ESSMAILI,M. FOZOUNIANDJ. LAALI
It follows that ρ♯ is a left A-module morphism. Hence, E is a φ-injective Banach
φ
left A-module. (cid:3)
Corollary 2.4. Let A be a commutative Banach algebra, and J be a closed ideal
of A such that φ 6= 0. Then A/J is φ-injective as a Banach left A-module.
|J
Proof. Since φ 6= 0, it is easy to check that (A/J)⊥∩(ker(φ))c 6= ∅. Now, apply
|J
(cid:3)
Theorem 2.3.
Corollary 2.5. Let A be a commutative Banach algebra, φ ∈ ∆(A) and let
E ∈ A-mod with I(φ,E) = {0}. Then for all ψ ∈ ∆(A)\{φ}, E ∈ A-mod is
ψ-injective.
Proof. Since φ 6= ψ there exists a ∈ A such that φ(a ) = 0 and ψ(a ) = 1. On
0 0 0
the other hand, since I(φ,E) = {0} we conclude that a ∈ E⊥ ∩ (ker(ψ))c and
0
(cid:3)
the proof is complete.
Now, we give some hereditary properties of φ-injectivity of Banach modules
that we shall use. Recall that E ∈ A-mod is faithful if A·x = {0} implies that
x = 0.
Theorem 2.6. Let A be a Banach algebra,E ∈ A-mod, φ ∈ ∆(A) and J be a
closed ideal of A such that φ 6= 0.
|J
(i) Suppose that J has an identity and E ∈ J-unmod. If E ∈ A-mod is
φ-injective, then E ∈ J-unmod is φ -injective.
|J
(ii) If E ∈ J-mod is φ -injective and faithful, then E ∈ A-mod is φ-
|J
injective.
Proof. (i) Suppose that E ∈ A-mod is φ-injective. Let F and K be in J-mod
and T : F −→ K be an admissible monomorphism with I(φ ,K) ⊆ ImT. We
|J
claim that the induced map, B(K,E) −→ B(F,E) defined by, R −→ R◦T is
J J
onto. Suppose that e is the identity of J. We can consider F and K as Banach
J
left A-modules with the following module actions:
a•f = (ae )·f (a ∈ A,f ∈ F),
J
a•k = (ae )·k (a ∈ A,k ∈ K).
J
We denote these A-modules with F and K, respectively. Take W ∈ B(F,E) and
J
define the map W : F −→ E by W(f) = W(f). For each a ∈ A and f ∈ F we
e e
have,
f e f
W(a•f) = W((ae )·f) = (ae )·W(f)
J J
= a·(e ·W(f)) = a·W(f)
f J
= a·W(f).
f
HEREDITARY PROPERTIES OF CHARACTER INJECTIVITY 5
So W is a left A-module morphism. Moreover, the map T : F −→ K defined by
T(f) = T(f) is an admissible monomorphism such that
f e e e
e I(φ,K) = span{a•k −φ(a)k : a ∈ A,k ∈ K}
= span{(ae )·k −φ(ae )k : a ∈ A,k ∈ K}
e J J
⊆ ImT = ImT.
Since E ∈ A-mod is φ-injective, theere exist S ∈ B(K,E) such that S◦T = W.
A
On the other hand, for each a ∈ J and k ∈ K we have
e e f
a·S(k) = S(a•k) = S((ae )·k) = S(a·k).
J
It follows that S ∈ B(K,E). Now, we conclude that E ∈ J-unmod is φ -
J |J
injective.
(ii) Let F and K be in A-mod and T : F −→ K be an admissible monomor-
phism and take W ∈ B(F,E). So W ∈ B(F,E) and there exists S ∈ B(K,E)
A J J
such that S ◦T = W. For each a ∈ J, b ∈ A and k ∈ K, we have
a·(S(b·k)−b·S(k)) = a·S(b·k)−(ab)·S(k)
= S(ab·k)−S(ab·k) = 0.
Since E ∈ J-mod is faithful, we conclude that S(b·k) = b·S(k). It follows that
S ∈ B(K,E) and the proof is complete. (cid:3)
A
Corollary 2.7. Let A be a Banach algebra, φ ∈ ∆(A) and J be a closed ideal of
A with an identity such that φ 6= 0. Then J ∈ A-mod is φ-injective if and only
|J
if J ∈ J-mod is φ -injective.
|J
B. E. Forrest in [5] introduced the notion of invariantly complemented sub-
modules in categories of Banach modules. In the sequel, we obtain some results
for φ-injectivity of invariantly complemented ideals.
Definition 2.8. ([5, Definition6.3])LetX beaBanachleftA-moduleandY bea
BanachA-submoduleofX. WesaythatY isleft (right) invariantly complemented
in X if there exists P ∈ B(X,Y) (P ∈ B (X,Y)) such that P2 = P and
A A
P(X) = Y.
Theorem 2.9. Let{E } be a collectionof BanachleftA-modules andconsider
α α∈Γ
E = ℓ1 − E as a Banach left A-module with the natural module action.
α∈Γ α
(i) If LE is φ-injective, then for each α ∈ Γ, E is φ-injective.
α
(ii) Conversely, if Γ is finite and each E is φ-injective, then E is φ-injective.
α
Proof. (i) It isobvious that each E isleft invariantly complemented inE. Hence,
α
for each α ∈ Γ, let P ∈ B(E,E ) such that P (E) = E and P2 = P . Also,
α A α α α α α
let i : E −→ E be the natural embedding of E into E.
α α α
Let E be φ-injective. Then there exists ρE ∈ B( B(A♯,E),E) such that is
φ A φ
a left inverse for ΠE : E −→ B(A♯,E). For each α ∈ Γ, we define the map
φ φ
ρα : B(A♯,E ) −→ E by
φ φ α α
ρα(T) = P ◦ ρE(i ◦T) (T ∈ B(A♯,E )).
φ α φ α φ α
6 M. ESSMAILI,M. FOZOUNIANDJ. LAALI
Weclaimthat ρα isaleftA-modulemorphismsuchthat ρα◦ Πα = I . Indeed,
φ φ φ Eα
since for each x ∈ E , i ◦ Πα(x) = ΠE(i (x)), so we have
α α φ φ α
ρα ◦ Πα(x) = P ◦ ρE(i ◦ Πα(x)) = P ◦ ρE( ΠE(i (x)))
φ φ α φ α φ α φ φ α
= P (i (x)) = x.
α α
Therefore, ρα◦ Πα = I . On the other hand, since P ∈ B(E,E ) it is easy
φ φ Eα α A α
to see that ρα is a left A-module morphism and the proof is complete.
φ
(ii) Suppose that for each α ∈ Γ, E is φ-injective. So, for each α ∈ Γ
α
there exists ρα ∈ B( B(A♯,E ),E ) for which ρα◦ Πα = I . Define the
φ A φ α α φ φ Eα
map ρ : B(A♯,E) −→ E by
φ
ρ(T) = ( ρα(P ◦T)) (T ∈ B(A♯,E)).
φ α α∈Γ φ
Since Γ is finite, ρ is well-defined. For each a ∈ A and T ∈ B(A♯,E) we have
φ
ρ(a·T) = ( ρα(P ◦(a·T))) = ( ρα(a·(P ◦T)))
φ α α∈Γ φ α α∈Γ
= (a· ρα(P ◦T)) = a·( ρα(P ◦T))
φ α α∈Γ φ α α∈Γ
= a·ρ(T).
Moreover, if x = (x ) is an arbitrary element of E, it is easy to see that
α α∈Γ
P ◦ ΠE(x) = Πα(x ). Hence,
α φ φ α
ρ◦ ΠE(x) = ( ρα(P ◦ ΠE(x))) = ( ρα( Πα(x )))
φ φ α φ α∈Γ φ φ α α∈Γ
= (x ) = x.
α α∈Γ
Therefore, we conclude that E is φ-injective. (cid:3)
Theorem 2.10. Let A be a Banach algebra, φ ∈ ∆(A), B be a subalgebra of A
and J be a closed left ideal of A. Then the following assertions holds:
(i) If B is left invariantly complemented in A and A is φ-injective in A-mod,
then B is φ-injective in A-mod.
(ii) If J is left invariantly complemented, then J and A/J are φ-injective in
A-mod if and only if A is φ-injective in A-mod.
Proof. (i) Since B is left invariantly complemented in A, there exists an onto
projection P ∈ B(A,B). Hence
A
∼
A = ImP ⊕kerP = B ⊕kerP,
as a Banach left A-module. Therefore, by Theorem 2.9 it follows that B is φ-
injective in A-mod.
(ii) Since J is a left invariant complemented ideal in A, there exists an onto
projection P ∈ B(A,J). We claim that A ∼= J ⊕ A as a Banach left A-module.
A J
To see this, define the map T : A −→ J ⊕ A by
J
T(a) = (P(a),a+J) (a ∈ A).
First, T is a left A-module morphism, because for each a,b ∈ A we have,
T(ab) = (P(ab),ab+J) = (aP(b),a(b+J))
= a·(P(b),b+J) = a·T(b).
HEREDITARY PROPERTIES OF CHARACTER INJECTIVITY 7
Ontheotherhand, ifa ∈ ImP ∩kerP,thenthereexistsb ∈ AsuchthatP(b) = a.
Hence, a = P(b) = P(P(b)) = P(a) = 0. This follows that ImP ∩kerP = {0}
and so T is one-to-one. Moreover, T is onto because for each (a,b+J) ∈ J ⊕ A
J
if we put c = a+b−P(b), then T(c) = (a,b+J). Now, the result follows from
(cid:3)
Theorem 2.9.
As an application of Theorem 2.10 and Corollary 2.4, we have the following
result for commutative Banach algebras.
Corollary 2.11. Let A be a commutative Banach algebra, φ ∈ ∆(A) and let J be
a closed invariant complemented ideal of A such that φ 6= 0. Then A ∈ A-mod
|J
is φ-injective if and only if J ∈ A-mod is φ-injective.
3. Applications to semigroup algebras
In this section, we apply our later results to study φ-injectivity of certain
commutativesemigroupalgebrasandgivesomeexamplesofnon-injectivemodules
which are φ-injective for each character φ. First, we need some basic facts about
semigroup algebras.
Let S be a semigroup and let E(S) = {s ∈ S : s2 = s}. We say that S is a
semilattice if S is commutative and E(S) = S. A semi-character on S is a non-
zero homomorphism φ : S −→ {z ∈ C : |z| ≤ 1}. The space of semi-characters on
S is denoted by Φ . The semi-character φ : S −→ {z ∈ C : |z| ≤ 1}, defined by
S b S
φ (t) = 1 b (t ∈ S),
S
is called the augmentation chbaracter on S. For each φ ∈ Φ , we associate the
S
map φ : ℓ1(S) −→ C defined by
b
1
φ(f) = φ(s)f(s) (f ∈ ℓ (S)).
Xs∈S
b
It is easy to verify that φ ∈ ∆(ℓ1(S)) and every character on ℓ1(S) arises in this
way. Indeed, we have
1
∆(ℓ (S)) = {φ : φ ∈ Φ }.
S
We also define the convolution of two elements f,g ∈ ℓ1(S) by
b
(f ∗g)(s) = f(u)g(v) (s ∈ S),
uXv=s
where f(u)g(v) = 0, when there are no elements u,v ∈ S with uv = s.
uv=s
Then (Pℓ1(S),∗,k·k ) becomes a Banach algebra that is called the semigroup
1
algebra of S. The following proposition immediately follows from Corollary 2.11.
Proposition3.1. LetS be a semilattice, φ ∈ ∆(ℓ1(S)) andI be a closedinvariant
complemented ideal in ℓ1(S) such that φ 6= 0. Then ℓ1(S) ∈ ℓ1(S)-mod is φ-
|I
injective if and only if I ∈ ℓ1(S)-mod is φ-injective.
Let ℓ1(N ) be the semigroup algebra on semigroup S = (N,∧) with the follow-
∧
ing product:
N×N −→ N, (m,n) −→ m∧n = min{m,n}.
8 M. ESSMAILI,M. FOZOUNIANDJ. LAALI
It is easy to check that Φ = {φ : n ∈ N}, where for each n ∈ N,
S n
φ (m) = b1 if m ≥ n (m ∈ N).
n (cid:26) 0 if m < n
b
Foreachn ∈ N,letI betheidealofℓ1(N )generatedbytheset{δ ,δ ,δ ,...,δ }.
n ∧ 1 2 3 n
It is easy to see that ℓ1(N )/I does not have an identity. So I does not have a
∧ n n
modular identity and using [12, Corollary 2.2.8 (ii)], we conclude that ℓ1(N )/I
∧ n
is not injective as a Banach left ℓ1(N )-module. Furthermore, we recall that
∧
ℓ1(N ) as a Banach left ℓ1(N )-module is not injective, because it does not have
∧ ∧
a right identity [2, Example 4.10]. As mentioned above, we regard the map
φ : ℓ1(N ) −→ C as a character on ℓ1(N ) which is defined by
n ∧ ∧
∞
φ (f) = f(i) (f ∈ ℓ1(N )).
n ∧
Xi=n
The following theorem shows that ℓ1(N ) as a left ℓ1(N )-module is φ-injective
∧ ∧
for each φ ∈ ∆(ℓ1(N )), although is not injective.
∧
Theorem 3.2. With the above notations, we have following assertions:
(i) ℓ1(N )/I as a Banach left ℓ1(N )-module is φ -injective, for each n ∈ N.
∧ n ∧ n
(ii) ℓ1(N ) as a Banach left ℓ1(N )-module is φ -injective, for each n ∈ N.
∧ ∧ n
Proof. (i)Sinceℓ1(N )iscommutative and(φ ) 6= 0, byCorollary2.4, itfollows
∧ n |In
that ℓ1(N )/I as a Banach left ℓ1(N )-module is φ -injective.
∧ n ∧ n
(ii) First, we show that for each n ∈ N, I is an invariant complemented ideal
n
of ℓ1(N ). To see this, suppose that the map P : ℓ1(N ) −→ I is defined by
∧ n ∧ n
n−1 ∞
P (f) = f(i)δ +( f(i))δ (f ∈ ℓ1(N )).
n i n ∧
Xi=1 Xi=n
It is easy to check that P is a projection on I . Moreover, if f or g belong to
n n
I we have P (f ∗g) = f ∗P (g). Now suppose that f,g are not in I . We can
n n n n
suppose that f = δ and g = δ such that n < i ≤ j. Hence, we have
i j
P (δ ∗δ ) = P (δ ) = δ = δ ∗δ = δ ∗P (δ ).
n i j n i n i n i n j
This follows that I is an invariant complemented closed ideal of ℓ1(N ).
n ∧
Case 1: We prove that A = ℓ1(N ) is φ -injective. Since I is invariantly
∧ 1 1
complemented in A, (φ ) 6= 0 and A is commutative, by Proposition 3.1, it
1 |I1
suffices to show that I ∈ A-mod is φ -injective.
1 1
Define the map ρ : B(A♯,I ) −→ I by
φ1 1 1
ρ(T) = T(e♯) (T ∈ B(A♯,I ).
φ1 1
Clearly ρ is a left inverse for Π♯ : I −→ B(A♯,I ). We claim that ρ is a left
φ1 1 φ1 1
A-module morphism.
For each f ∈ A and g ∈ I we note that f ∗ g = φ (f)g. Hence, if T ∈
1 1
B(A♯,I ), then for each f,g ∈ A we have
φ1 1
T(f ∗g −φ (g)f) = f ·T(g −φ (g)e♯).
1 1
HEREDITARY PROPERTIES OF CHARACTER INJECTIVITY 9
In the case where g = δ , we conclude that
1
T(φ (f)δ −φ (g)f) = T(f ∗g −φ (g)f)
1 1 1 1
= f ·T(g −φ (g)e♯)
1
= f ∗T(g −φ (g)e♯)
1
= φ (f)T(δ −φ (δ )e♯)
1 1 1 1
= φ (f)T(δ −e♯).
1 1
Therefore, T(f) = φ (f)T(e♯). This follows that for each f ∈ A and T ∈
1
B(A♯,I ) we have
φ1 1
ρ(f ·T) = (f ·T)(e♯) = T(e♯f) = T(f)
= φ (f)T(e♯) = f ·T(e♯) = f ·ρ(T).
1
Hence, ρ is a left A-module morphism which is also a left inverse for Π♯. There-
φ1
fore, ℓ1(N ) is φ -injective as a Banach left ℓ1(N )-module.
∧ 1 ∧
Case 2: We prove that ℓ1(N ) is φ -injective as a Banach left ℓ1(N )-module
∧ n ∧
for each n ≥ 2. Since I ∈ I -mod has the identity δ , so it is faithful. By
n n n
Theorem 2.6, it suffices to show that I ∈ I -mod is (φ ) -injective. Let φ
n n n |In n
be the semi-character on N associated with φ . Since S = ({1,2,3,...,n},∧)
∧ n
c
is a semilattice and φ (1) = 0, it follows from [4, Theorem 2.1] that I =
n n
ℓ1({1,2,3,...,n},∧) is (φ ) -amenable. Therefore, we conclude that c (S) is
c n |In 0
(φ ) -flat in mod-I by [10, Proposition 3.1]. Hence, I = c (S)∗ is (φ ) -
n |In n n 0 n |In
injective in I -mod and the proof is complete. (cid:3)
n
Let ℓ1(N ) be the semigroup algebra on semigroup S = (N,∨) with the follow-
∨
ing product:
N×N −→ N, (m,n) −→ m∨n = max{m,n}.
It is easy to check that Φ = {ψ : n ∈ N}∪{ψ }, where ψ is the augmentation
S n S S
character and for each n ∈ N,
b c c
1 if m ≤ n
ψ (m) = (m ∈ N).
n (cid:26) 0 if m > n
b
In [11, Example 5.6], it is proved that ℓ1(N ) is not injective as a Banach left
∨
ℓ1(N )-module. In the following theorem, we show that ℓ1(N ) is φ-injective for
∨ ∨
each φ ∈ ∆(ℓ1(N )).
∨
Theorem 3.3. ℓ1(N ) as a Banach left ℓ1(N )-module is φ-injective for each
∨ ∨
φ ∈ ∆(ℓ1(N )).
∨
Proof. By [4, Corollary 2.2], it follows that ℓ1(N ) is character amenable . Hence,
∨
for each φ ∈ ∆(ℓ1(N )), ℓ1(N ) is φ-amenable. On the other hand, since N is
∨ ∨ ∨
weakly cancellative by [2, Theorem 4.6], we conclude that c (N ) is a Banach
0 ∨
ℓ1(N )-module . Hence, c (N ) is φ-flat as a Banach right ℓ1(N )-module [10,
∨ 0 ∨ ∨
Proposition 3.1]. This follows that c (N )∗ = ℓ1(N ) is φ-injective as a Banach
0 ∨ ∨
left ℓ1(N )-module. (cid:3)
∨
10 M. ESSMAILI,M. FOZOUNIANDJ. LAALI
For each n ∈ N, let J be the closed ideal of A = ℓ1(N ) generated by
n ∨
{δ ,δ ,...}. It is easy to see that J is invariantly complemented in ℓ1(N ).
n n+1 n ∨
Indeed, it is sufficient to consider the map Q : A −→ J defined by
n n
n ∞
Q (f) = ( f(i))δ + f(i)δ (f ∈ A).
n n i
Xi=1 i=Xn+1
It is straightforward to check that Q is an onto projection in B(A,J ).
n A n
As a consequence of Corollary 2.11 and Theorem 3.3, we give the following
result.
Corollary 3.4. For each n ∈ N, J as a Banach left ℓ1(N )-module is φ-injective
n ∨
for each φ ∈ ∆(ℓ1(N )) with φ 6= 0.
∨ |Jn
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Faculty of Mathematical and Computer Science, Kharazmi University, 50
Taleghani Avenue, 15618 Tehran, Iran.
E-mail address: [email protected]
E-mail address: [email protected]
E-mail address: [email protected]